Subcategories

Construction 3.1.1. For categories \(C\), \(D\) and \(E\), we construct a map \[ \Map (C,D) \to \Map (\Map (E,C), \Map (E,D)) \] by currying the internal composition functor from Proposition 1.4.8: \[ \mu _{E,C,D}\colon \Map (C,D)\times \Map (E,C)\longrightarrow \Map (E,D). \] For an absolute functor \(f\colon C\to D\), the specialization of the resulting map at \(f\) is identified with the postcomposition functor \(f_*\colon \Map (E,C)\to \Map (E,D)\) constructed in Construction 1.4.3. This follows by uncurrying and comparing the two double-evaluation functors. Here we use the convention from Construction 1.4.1 of passing back and forth between functors and their representing points in the mapping anima.

Definition 3.1.2 (Subcategory). A functor \(f\colon A \to C\) is called a subcategory if the following two conditions are satisfied:

(1)

The map \(f_*\colon \Map ([1],A) \to \Map ([1],C)\) is an embedding;

(2)

For every category \(D\), the induced square

is a pullback square. Here the two vertical maps are given by Construction 3.1.1.

Remark 3.1.3. The above notion of subcategory corresponds to what is known as a replete subcategory in the higher category theory literature. There is no sensible notion of non-replete subcategories in the synthetic setting.

Example 3.1.4. The identity functor \(\id _C\colon C \to C\) and the empty category \(\emptyset \to C\) are both seen to be subcategories.

Lemma 3.1.5. Every subcategory \(f\colon A \to C\) is an embedding.

Definition 3.1.6 (Collection of morphisms). Let \(C\) be a category. A collection of morphisms in \(C\) is an anima \(M\) equipped with an embedding \[ m\colon M \hookrightarrow \Map ([1],C). \] We say that \(M\) is closed under composition if the following two conditions are satisfied:

• If a morphism \(f\colon x \to y\) in \(C\) is in \(M\), then so are \(\id _x\) and \(\id _y\). More concretely, there are commutative diagrams of the following form:

  

• For maps \(f\colon x \to y\) and \(g\colon y \to z\) in \(M\), also the composition of \(f\) and \(g\) is in \(M\). More concretely, there exists a dotted arrow in the following diagram:

  

Convention 3.1.7. Let \(m\colon M \hookrightarrow \Map ([1],C)\) be a collection of morphisms in \(C\). We say that a morphism \(f\colon x \to y\) in \(C\) is contained in \(M\), or simply is in \(M\), if the corresponding absolute object of \(\Map ([1],C)\) factors through \(m\). Since \(m\) is an embedding, such a factorization, if it exists, is unique, so being contained in \(M\) is a property. Thus, in phrases such as “a morphism in \(M\)”, the word “morphism” refers to a morphism in \(C\) contained in the subanima \(M \hookrightarrow \Map ([1],C)\), not to a morphism of the anima \(M\).

Example 3.1.8. Taking \(M = \Map ([1],C)\), it is clear that the identity map \(M \to \Map ([1],C)\) is a collection of morphisms in \(C\) closed under composition.

Example 3.1.9. Let \(A \to C\) be a subcategory. Then the map \(M_A := \Map ([1],A) \to \Map ([1],C)\) is a collection of morphisms in \(C\) closed under composition. This is immediate from the fact that the functor \(A \to C\) preserves compositions and identities.

Lemma 3.1.10. For a category \(C\), the map \(\pi _{\Iso }^{\simeq }\colon \Iso (C)^{\simeq } \to \Map ([1],C)\) defines a collection of morphisms in \(C\) closed under composition.

Corollary 3.1.11. In the situation of Axiom H, the functor \(\Map ([1],\lra {M}_C) \to M\) is an equivalence.

Corollary 3.1.12. In the situation of Axiom H, the functor \(i_{M}\colon \lra {M}_C \to C\) exhibits \(\lra {M}_C\) as a subcategory of \(C\). In particular, \(i_M\) is an embedding.

Lemma 3.1.13. Let \(f\colon A \to C\) be a subcategory and let \(M_A := \Map ([1],A) \hookrightarrow \Map ([1],C)\) denote the induced collection of morphisms in \(C\). Then the map \(i_{M_A}\colon \lra {M_A}_C \to C\) factors through an equivalence \(\lra {M_A}_C \iso A\).

Corollary 3.1.14. When \(M = \Map ([1],C)\) is the collection of all morphisms, the map \(i_M\colon \lra {M}_C \to C\) is an equivalence. □

Definition 3.1.15. Let \(C\) and \(D\) be categories, and assume that they come equipped with collections of morphisms \(m\colon M \hookrightarrow \Map ([1],C)\) and \(n\colon N \hookrightarrow \Map ([1],D)\). A functor \(f\colon C \to D\) is said to send morphisms in \(M\) to morphisms in \(N\) if there exists a map \(\tilde {f}\colon M \to N\) making the following diagram commute:

Since the map \(n\) is an embedding, the map \(\tilde {f}\colon M \to N\) is unique if it exists.

Construction 3.1.16. Let \(C\) and \(D\) be categories equipped with collections of morphisms \(M\) and \(N\) closed under composition, and let \(f\colon C \to D\) be a functor which sends morphisms in \(M\) to morphisms in \(N\). We construct a functor \(f\vert \colon \lra {M}_C \to \lra {N}_D\) making the following diagram commute:

By the defining property of \(\lra {N}_D\), it suffices to show that the map \(f \circ i_M\colon \lra {M}_C \to D\) sends all morphisms to morphisms in \(N\). But this follows from the following commutative diagram:

Remark 3.1.17 (Exercise 3.8.3). The construction \(f \mapsto f\vert \) satisfies the following properties:

(1)

The map \(f\vert \) is uniquely determined by \(f\).

(2)

Given an identification \(f = f'\), we obtain an induced identification \(f\vert = f'\vert \).

(3)

The construction is compatible with identities and composition through identifications \[ (\id _C)\vert = \id _{\lra {M}_C} \qquadtext {and}\qquad (g \circ f)\vert = g\vert \circ f \vert . \]

(4)

If the maps \(f\) and \(\tilde {f}\) are equivalences, then also \(f\vert \colon \lra {M}_C \to \lra {N}_D\) is an equivalence.